single sample
statistical error that occurs when a researcher rejects the null hypothesis when it is actually true, such as falsely claiming cafe wait times have increased
Type I Error (alpha)
Which scenario is appropriate for a two-sample $t$-test: (1) Testing two diets on a herd of cows split evenly into two distinct groups, or (2) Giving cows Diet A for 3 weeks and then Diet B for 3 weeks?
Scenario 1, because it consists of two independent groups (Scenario 2 lacks independence)
How can you recognize an experiment as a paired design rather than an independent two-sample design?
Observations are naturally matched or paired (e.g., twin pairs, before/after measurements on the same subject), and analysis is conducted on the individual differences.
What is the typical null hypothesis (H_0) when running a Chi-square (chi^2) test of independence on a contingency table?
The two categorical variables are independent
In a chi^2 Goodness of Fit test with k categories, what is the formula for degrees of freedom?
df = k - 1
When testing a single population mean where the population standard deviation ($\sigma$) is known, which test statistic is used
One-sample Z-test statistic
When assuming two independent normal populations have unknown but equal variances, what estimate is used to combine s_1^2 and s_2^2 in the denominator of the test statistic
The Pooled Variance
(sp2) =( (n1 - 1)s12 + (n2 - 1)s22^2)/(n1 + n2 - 2)
In a paired t-test examining whether a quiet room yields higher concentration scores than a noisy lounge, what is the null hypothesis (H_0)
H_0: mu_d = 0 (difference is nothing)
What is the formula for degrees of freedom ($df$) in a Chi-square test of independence for a contingency table?
df = (r - 1)(c - 1)
What rule must be satisfied regarding expected frequencies for a chi^2 Goodness of Fit test to be reliable?
All expected frequencies should be > 1, and at least 80% of expected frequencies should be > 5
average cafe wait time is normally distributed with mu = 4.5 minutes and sigma = 1.2 minutes. Staff sample n = 36 customers and find bar{x} = 4.8 minutes. What is the computed test statistic Z_obs?
Z_{obs} = 1.5
A study compares weight loss from Diet A (n1 = 15) and Diet B (n2 = 12). What are the degrees of freedom for a two-sample pooled t-test
df = 25
In a study of n = 30 twin pairs comparing quiet vs. noisy study environments, the sum of differences is 138. What is the sample mean difference x_bar_d?
x_bar_d = 4.6 = (38/30)
Among 100 students (60 Arts, 40 Science; 55 planning grad school, 45 not), what is the expected count (E) of Arts students planning on a graduate degree under H_0?
E = 33 = (60 x 55)/100 = (r total)(c total)/total
A survey asks 150 students if they prefer cats or dogs. Under H_0 that preferences are equally split, what are the expected counts for cats and dogs?
75 Cats and 75 Dogs (150 x 0.5 = 75)
A quality team tests n = 100 smartphones to see if battery life exceeds mu = 14 hours. The sample mean is x_bar = 14.6 hours with s = 2.8 hours. Calculate the test stat
t_obs = approx 2.14
In Suburb 1 (n_1 = 30), 19 hummingbirds have beaks >7 cm. In Suburb 2 (n_2 = 30), 3 have beaks >7 cm. What is the pooled sample proportion p_hat?
p_hat = 0.3667, (19 + 3)/(30+30)
For n = 30 paired differences with sample standard deviation s_d = 6.53, what is the standard error of the mean difference SE(x_bar_d)?
SE = 1.19 = 6.53/sqrt(30)
In a study on drink type vs. performance (N = 160), Energy Drink x Good Performance has an observed count of O = 54 and expected count of E = 46.5. Calculate its contribution to the chi^2 statistic
approx 1.21 = (54 - 46.5)^2/46.5
In the cat vs. dog survey (N = 150, Expected = 75 each), observed counts are 85 for cats and 65 for dogs. Compute the chi^2 test statistic.
chi^2_obs = 2.67 = (85 - 75)^2/75 + (65 - 75)^2/75
A game claims 90% (p_0 = 0.90) of matches occur within 30 seconds. In n = 180 random matches, 156 were completed within 30 seconds (x = 156). Using the normal approximation to the binomial, calculate Z_obs.
Z_obs = approx -1.49
mu = 180 x 0.9 = 162; sigma = sqrt(180 x 0.9 x 0.1} = approx 4.025; Z = (156 - 162)/4.025 = -1.49
Orange trees given Fertilizer A (n_1 = 11) had x_bar_1 = 14, s_1 = 5. Trees given Fertilizer B (n_2 = 11) had x_bar_2 = 21.18, s_2 = 6.79. Test for a difference
s_p^2 =35.58, t_stat = (mu_1 - mu_2) / sqrt(s2p * (1/n1 + 1/n2)) = -2.82359, reject
Given x_bar_d = 4.6 and SE(x_bar_d) = 1.19 for n = 30 pairs, calculate the test statistic and state its degrees of freedom, and find the p=value
t_obs = 3.86 with df = 29 (4.6/1.19), (30-1)
A survey of 400 students across 4 years of study and 4 lunch preferences yields a test statistic X^2 = 38.286. State the degrees of freedom (df) and decision at alpha = 0.05
df = 9; Reject H_0 (conclude food preference depends on year of study since p = 1.55 x 10^-5 < 0.05
Lunch distribution on South Lawn is claimed to be 20% Science, 70% Arts, and 10% Other (N = 400). Observed counts are Science = 70, Arts = 300, Other = 30. Calculate the expected counts and the total chi^2 test statistic. Make a decision
Expected: Science = 80, Arts = 280, Other = 40, chi^2_obs = 5.18, p-value = 0.07507